A note on the kissing numbers of superballs
type: Item
instance of: scholarly article
title: A note on the kissing numbers of superballs
zbMATH Open document ID: 1118.52009
author: Lanju Xu
published in: Discrete \& Computational Geometry
publication date: 2007-04-26T00:00:00Z
review text: For an \(n\)-dimensional convex body \(K\subset{\mathbb R}^n\), the largest number of non-overlapping translates of \(K\) such that \(K\) has a common (boundary) point with each translate is called the translative kissing number of \(K\) and is denoted by \(N(K)\). Furthermore, for \(\alpha\geq 1\) the \(n\)-dimensional superball is defined as \({\mathcal B}_{\alpha}=\left\{(x_1,\dots,x_n)\in{\mathbb R}^n:\bigl(\sum_{i=1}^n| x_i| ^{\alpha}\bigr)^{1/\alpha}\leq 1\right\}\). In [Discrete Comput. Geom. 21, 233--242 (1999; Zbl 0924.52011)], \textit{D. E. Larman} and \textit{C. Zong} determined the translative kissing number for different polyhedra and gave an exponential lower bound for \(n\)-dimensional superballs. In the present paper the author provides an improvement of this lower bound, showing that \((1/n)\log_2N({\mathcal B}_{\alpha})\geq 1+o(1)\) as \(\alpha\) goes to infinity. Moreover, lower bounds are obtained based on binary codes which are constructive in exponential and polynomial time.
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 5146538
zbMATH Keywords: kissing number
zbMATH Keywords: superballs
reviewed by: Maria A. Hernández Cifre
MaRDI profile type: Publication
full work available at URL: s00454-006-1256-9
OpenAlex ID: W2064139110
DOI: 10.1007/S00454-006-1256-9
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description: scientific article; zbMATH DE number 5146538